An Approach to Qualitative Belief Change Modulo Ontic Strength

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An Approach to Qualitative Belief Change Modulo Ontic Strength

Gavin Rens

¹

and Gabriele Kern-Isberner

²

Abstract. Sometimes, strictly choosing between belief revision and belief update is inadequate in a dynamical, uncertain environment.

Boutilier combined the two notions to allow updates in response to external changes to inform an agent about its prior beliefs. His approach is based on ranking functions. Rens proposed a new method to trade off probabilistic revision and update, in proportion to the agent’s confidence for whether to revise or update. In this paper, we translate Rens’s approach from a probabilistic setting to a setting with ranking functions. Given the translation, we are able to compare Boutilier’s and Rens’s approaches. We found that Rens’s approach is an extension of Boutilier’s.

1 Introduction

Traditionally,belief revisionis regarded as change of beliefs about the objective static state of the world. Andbelief updateis regarded as change of beliefs due to recording the change which occurred in the underlying state of the world. We shall use the generic termbelief changeto including belief update and belief revision. An agent may not always be certain whether an observation is a side-effect of an action/event (requiring update), or whether the observation did not have a physical cause and is thus pure information (requiring revision).

Boutilier [3] proposed a generalized qualitative update procedure, which combines both belief update and revision. He used ranking functions as advocated by Spohn [13, 14] to capture notions of preference. Rens [11] proposed a quantitative approach to mix probabilistic belief update and revision, where the trade-off is controlled by the so-calledontic strengthof the observation received. To our knowledge, his “mixture” method is novel. In this paper, we propose a translation of Rens’s method back to a qualitative setting using Spohn-rankings. The difference in the present approach to that of Boutilier is that ours trades belief update and revision off in proportion to the agent’s judgement of the ontic strength of the received evidence.

There are several reason why we would like a qualitative version of Rens’s hybrid stochastic belief change (HSBC).

• Ordering preferred worlds by ranking them instead of providing exact probabilities may be more intuitive for agent designers.

• Some domains may not require the agent to work with precise values like probabilities, and computations over ranked preferences are then cheaper, because finding the minimum of a set is generally cheaper than finding its sum (the distinction between mini- mization and summation will become clear later).

1Centre for AI Research, and University of KwaZulu-Natal, School of Math- ematics, Statistics and Computer Science, and CSIR Meraka, South Africa

2Dortmund University of Technology, Dortmund, Germany

• We may gain insights about the relationship between belief revision and belief update when analysed in the qualitative belief change setting.

LetLbe the classical propositional language, andW the (finite) set of possible worlds (valuations) induced from a finite set of propositional variables. We denote the models of a sentenceα ∈ Lby JαKand the fact thatwsatisfiedαbywα. For a set of sentences K ⊆ L,JKK := {w ∈ W | ∀β ∈ K, w β}. We refer to a probability function or a ranking function as anepistemic state. In this paper, we denote the result of a belief change operation as◦α, whereis an epistemic state and◦is the operator. If we need to refer to the value of a particular worldwin the changed epistemic state, we write^◦_α(w).

Next, we review the essentials of Rens’s HSBC construction.

In Section 3, we provide the qualitative, rank-based translation of HSBC (i.e., HQBC). We analyse our HQBC with respect to two fundamental classical rationality postulates in Section 4. In Section 5, we compare our hybrid qualitative belief change construction to Boutilier’sgeneralized updateconstruction. Two examples are presented in Section 6, and we end the paper with a summary of what has been achieved here, and a discussion about related and future work.

2 Hybrid Belief Change via Probability Theory

Rens [11] proposed thehybrid stochastic belief change(HSBC) operation to combine notions of probabilistic belief revision and probabilistic belief update. HSBC may be employed in agents who deal with uncertainty by maintaining a probability distributionbover possible worldswthey could be in. That is,b:W →[0,1], such that P

w∈Wb(w) = 1, andb(α) :=P

w∈W,wαb(w)for allα∈L.b will often be represented as a set of pairs{(w, p) | w ∈ W, p ∈ [0,1]}. We refer to bas an epistemic state in the context of this work. In the HSBC framework, an agent maintains an epistemic state, which changes as new information is received or observed.

Rens [11] proposes the tuplehW,Evt, T, E, O,osito formalize the HSBC framework, where

• Wis a set of possible worlds;

• Evtis a set of atomic events;

• T : W ×Evt×W → [0,1]is atransition functionsuch that for everye ∈ Evt andw ∈W,P

w⁰∈WT(w, e, w⁰) = 1, and T(w, e, w⁰) models the probability of a transition to worldw⁰, given the occurrence of eventein worldw;

• Eis theevent functionsuch thatE(e, w) =P(e|w), the probability of the occurrence of eventeinw;

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• O : L×W → [0,1]is anobservation functionsuch that for every worldw,P

α∈ΩO(α, w) = 1, andO(α, w) models the probability of observingαinw, whereΩ ⊂Lis the set of possible observations, up to equivalence, and where ifα ≡β, then O(α, w) =O(β, w), for all worldsw;³

• os : Ω×W → [0,1]such thatos(α, w)is the agent’sontic strengthforαperceived inw.

In HSBC, the epistemic state updated withα(denotedbα) is defined as

bα:=

(w⁰, p⁰)|w⁰∈W, p⁰= 1

γO(α, w⁰) X

w∈W

X

e∈Evt

T(w, e, w⁰)E(e, w)b(w) ,

whereγis a normalizing factor.

It is mostly agreed upon that Bayesian conditioning corresponds to classical beliefexpansion. This is evidenced by Bayesian conditioning (BC) being defined only whenb(α)6= 0(i.e., whenαdoes not contradict the agent’s current beliefs). In other words, one could define revision to be

bBCα:={(w, p)|w∈W, p=P(w|α)}, as long asP(α)6= 0, where

P(w|α) := O(α, w)b(w) P

w⁰∈WO(α, w⁰)b(w⁰).⁴ (1) To accommodate cases whereb(α) = 0, that is, whereαcontra- dicts the agent’s current beliefs and its beliefs need to be revised in the stronger sense, we shall make use ofimaging. Imaging was in- troduced by Lewis [9] as a means of revising a probability function.

Informally, Lewis’s original solution for accommodating contradicting evidenceαis to move the probability of each world to its closest, α-world. Lewis made the strong assumption that every world has auniqueclosestα-world. More general versions of imaging allow worlds to haveseveral, equally proximate closest worlds.

In two papers, Rens [11] and colleagues [12] proposegeneralized imaging: LetMin(α, w, d)be the set ofα-worlds closest towmea- sured withd, some acceptable measure of distance between worlds (e.g., Hamming or Dalal distance). Formally,

Min(α, w, d) :={w⁰∈JαK| ∀w⁰⁰∈JαK, d(w, w⁰)≤d(w, w⁰⁰)}.

Then generalized imaging (denotedGI) is defined as bGIα:=

(w, p)|w∈W, p= 0ifw6∈JαK,

elsep= X

w⁰∈W w∈Min(α,w⁰,d)

b(w⁰)/|Min(α, w⁰, d)| .

Rens [11] argues that if observation likelihoods are known, they should be used to weight the probabilities computed by theGIoper- ation; a new imaging operation is thus defined as

bOGIα:=n

(w, p)|w∈W, p= O(α, w)b^GI_α(w) P

w⁰∈WO(α, w⁰)b^GIα(w⁰) o

, where the denominator is a normalizing factor. At last, with respect to revision, Rens [11] definesBCIto revise by conditioning when the

3≡denotes logical equivalence.

4Note thatb(α)is equivalent toP(α).

evidence does not contradict the agent’s current beliefs, and to revise by imaging otherwise:

bBCIα:=

bBCα ifb(α)>0 bOGIα ifb(α) = 0

Finally, Rens [11] proposes a way of trading off the probabilistic update and probabilistic revision, using the notion of ontic strength.

He argues that an agent could reason with a range of degrees for information being ontic (the effect of a physical action or occurrence) or epistemic (purely informative). It is assumed that the higher the information’s degree of being ontic, the lower the epistemic status of that information. “An agent has a certain sense of the degree to which a piece of received information is due to a physical action or event in the world. This sense may come about due to a combination of sensor readings and reasoning. If the agent performs an action and a change in the local environment matches the expected effect of the action, it can be quite certain that the effect is ontic information,” [11, p. 129].

os(α, w)is defined to equal 1 whenαis certainly ontic inw, and 0 whenαis certainly epistemic (the epistemic strength ofαinwis es(α, w) := 1−os(α, w)).

The hybrid stochastic change of epistemic statebdue to new in- formationαwith ontic strength (denotedbα) is defined as

bα:=n

(w, p)|w∈W, p= 1

γ⁰

es(α, w)b^BCIα (w) +os(α, w)bα(w)o , whereγ⁰is a normalizing factor so thatP

w∈Wbα(w) = 1.

3 Hybrid Belief Change via Ranking Theory

Letκbe a ranking on worlds inW, representing the agent’s current epistemic state, as first proposed by Spohn [13]. That is,κ : W → N∪ {∞}, whereN = {0,1,2, . . .}, such that there exists aw ∈W for whichκ(w) = 0andκ(wi) ≤κ(wj)is interpreted as worldwi being at least as plausible or preferred as worldwj. κ(w^×) =∞is meant to indicate thatw^×is impossible, implausible, least preferred. Worldsw⁰for whichκ(w⁰) = 0are considered most plausible, most preferred, or believed. In fact, ranking functions are rankings ofimplausibility. The degree of plausibility of proposition αis

κ(α) := min

w∈W,wα{κ(w)}. (2)

We shall denote an agent’s belief set, given epistemic stateκ, as Bel(κ) :={β∈L|κ⁻¹(0)⊆JβK}.

Since Spohn’s ranking functions can be considered as the loga- rithm of probabilities [7], when translating from probability theory to Spohn ranking theory, multiplication becomes addition, division becomes subtraction, and summation (P

w∈W) becomes minimiza- tion (minw∈W).

Conditional plausibility is defined as [13]

κ(β|α) :=κ(α∧β)−κ(α).

Letφwbe a complete theory forw. It will be useful to know that κ(α∧β) = κ(α | β) +κ(β). And consequently, thatκ(α) = minw∈W{κ(α | φw) +κ(w)}. One can also defineκ(w | α)in terms ofκ(α|w)as follows.

κ(w|α) = κ(φw∧α)−κ(α)

= κ(α|φw) +κ(w)−κ(α).

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A direct translation of the Bayes Rule would suggest that the above result is analogous to that rule.

Definition 1. The tuplehW,Evt, TQ, EQ, OQ,osiis a hybrid qualitative belief change (HQBC) model, where

• Evtis a set of atomic events;

• TQ:W×Evt×W →N∪ {∞}is atransition rankingsuch that for everyw ∈ W ande ∈ Evt,minw⁰∈W{TQ(w, e, w⁰)} = 0 andTQ(w, e, w⁰)models the plausibility of a transition to world w⁰, given the occurrence of eventein worldw;

• EQ :Evt ×W → N∪ {∞}is theevent rankingsuch that for everyw∈W,mine∈Evt{EQ(e, w)}= 0andEQ(e, w)models the plausibility of the occurrence of eventeinw;

• OQ:L×W →N∪ {∞}is anobservation rankingsuch that for every worldw,minα∈L{O(α, w)} = 0andOQ(α, w)models the plausibility of observingαin wand where ifα ≡ β, then OQ(α, w) =OQ(β, w), for all worldsw⁵;⁶

• os :L×W →N, whereos(α, w)is the agent’sontic strength forαperceived inw(note thatosis not aκ-function).

In the qualitative version of the hybrid belief change framework, epistemic strength is defined as the complement of ontic strength.

Unfortunately, the notion of complement is not strictly defined for ranking theory. We thus define epistemic strength as the complement of ontic strength with respect to a ‘top’ value.

Definition 2. Letτ be an even number inN, but do not letτ =∞.

Epistemic strength is defined as theτ-complement ofos.

es(α, w) :=τ−os(α, w).

for all possible observationsαand for all worldsw∈W.

To specify that the agent has no preference for an observation being ontic or epistemic, chooseos(α, w) = τ /2for allw. Then es(α, w) =τ /2for allw.⁷

Letκbe regarded as an agent’s epistemic state andαa new piece of information to be accommodated. We can define the operation which revises an epistemic state using conditional plausibility:

κCPα=:{(w, n)|w∈W, n=Q(w|α)}, as long asκ(α)6=∞, where

Q(w|α) :=OQ(α, w) +κ(w)− min

w⁰∈W{OQ(α, w⁰) +κ(w⁰)}

is justified by the translation ofP(w |α)(Eq. 1) from probability theory to ranking theory. The definition ofQ(w | α) can also be derived from first principles, which we leave out here.

As with probabilistic conditionalization, plausibilistic conditionalization is undefined when the evidence/observation is inconsistent with the agent’s current epistemic state. A plausibilistic version of imaging can deal with this problem in the qualitative setting:

TranslatebGIαto κGIα:=n

(w, n)|w∈W, n=∞ifw6∈JαK, elsen= min

w⁰∈W w∈Min(α,w⁰,d)

{κ(w⁰)}o .

7Due toτbeing even,τ /2is guaranteed to be a whole number.

Example 1. Let the vocabulary be {q, r, s} and the current epistemic state κ1 = {(¯q¯r¯s,0), (¯qr¯s,1), (q¯r¯s,2), (qr¯s,3), (qrs,∞),(q¯rs,∞),(¯qrs,∞),(¯q¯rs,∞)}. Letdbe defined as Ham- ming distance. Suppose the observationαreceived is(q∧r)∨(q∧

¬r∧s). Then

Min(α, qrs, d) ={qrs} Min(α, qr¯s, d) ={qr¯s}

Min(α, q¯rs, d) ={qrs}¯ Min(α, q¯r¯s, d) ={qr¯s, q¯rs}

Min(α,qrs, d) =¯ {qrs} Min(α,qr¯¯s, d) ={qr¯s}

Min(α,q¯¯rs, d) ={qrs}¯ Min(α,q¯¯r¯s, d) ={qr¯s, q¯rs}

and

• κ^GI_1α(qrs) = min{κ(qrs), κ(¯qrs)}= min{∞,∞}=∞,

• κ^GI_1α(qr¯s) = min{κ(qr¯s),κ(qr¯¯s),κ(¯qr¯s),κ(¯q¯rs)}¯

= min{3,2,1,0}= 0,

• κ^GI1α(qrs) = min{κ(q¯ rs),¯ κ(qr¯¯s),κ(¯q¯rs),κ(¯q¯rs)}¯

= min{∞,2,∞,0}= 0,

• κ^GI_1α(qr¯¯s) =κ^GI_1α(¯qrs) =κ^GI_1α(¯qr¯s) =κ^GI_1α(¯qrs) =¯ κ^GI_1α(¯q¯r¯s)

=∞.

Notice that(q∧r)∨(q∧ ¬r∧s)does not contradictκ1 (i.e., κ1((q∧r)∨(q∧ ¬r∧s))6=∞). To show that qualitative imaging can deal with observations contradicting the agent’s epistemic state, consider the following example.

Example 2. We consider the same setting as in Example 1. Suppose the observationβreceived isq∧s. Note thatκ1(β) = ∞, that is, q∧sis deemed impossible inκ1. Then

• κ^GI_1β(qrs) = min{κ(qrs), κ(qr¯s), κ(¯qrs), κ(¯qr¯s)} = min{∞,3,∞,1}= 1,

• κ^GI_1β(q¯rs) = min{κ(q¯rs),κ(q¯r¯s),κ(¯q¯rs),κ(¯q¯r¯s)}

= min{∞,2,∞,0}= 0,

• κ^GI_1β(qr¯s) =κ^GI_1β(qr¯¯s) =κ^GI_1β(¯qrs) =κ^GI_1β(¯qr¯s) =κ^GI_1β(¯q¯rs) = κ^GI_1β(¯q¯r¯s) =∞.

Now, qualitative generalized imaging can be weighted/modulated by the plausibility of the evidence in a particular world:

κOGIα:={(w, n)|w∈W, n=κ^GI_α(w) +OQ(α, w)−δ}, whereδis a normalization factor defined as

δ:= min

w∈W{κ^GIα(w) +OQ(α, w)}.

Example 3. Continuing with the previous examples, suppose OQ(q∧s,q¯r¯¯s) = 1 and for allw 6= ¯qr¯¯s, OQ(q∧s, w) = 0.

Then

• κ^OGI_1β(qrs) = min{κ(qrs) + 0, κ(qr¯s) + 0, κ(¯qrs) + 0, κ(¯qr¯s) + 0} −δ= min{∞+ 0,3 + 0,∞+ 0,1 + 0} −δ= 1−1 = 0,

• κ^OGI_1β(q¯rs) = min{κ(q¯rs) + 0,κ(q¯r¯s) + 0,κ(¯q¯rs) + 0,κ(¯q¯r¯s) + 1} −δ= min{∞+ 0,2 + 0,∞+ 0,0 + 1} −δ= 1−1 = 0,

• κÔGI_1β(qr¯s) = κÔGI_1β (q¯r¯s) = κÔGI_1β(¯qrs) = κÔGI_1β(¯qr¯s) = κÔGI_1β(¯q¯rs) =κÔGI_1β(¯qr¯¯s) =∞.

In Example 2,qrs¯ is most plausible inκ^GI_1β, but in Example 3, due toq∧sbeing slightly less plausibly perceived inq¯¯r¯sthan in any other world,q¯¯r¯s becomes slightly less plausible inκ^OGI_1β. Thus, in

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Example 3,qrsandq¯rsshare the status of being most plausible in the agent’s revised epistemic state.

Finally, a qualitative version ofBCIcan be defined, which revises by conditional plausibility when the evidence does not contradict the agent’s current beliefs, and revises by qualitative imaging otherwise:

κCPIα:=

κCPα ifκ(α)6=∞ κOGIα ifκ(α) =∞ Turning now to belief update,b αis translated to κ α:=n

(w⁰, n)|w⁰∈W, n= (3)

OQ(α, w⁰) + min

w∈W e∈Evt

TQ(w, e, w⁰) +EQ(e, w) +κ(w) −δ⁰ o

,

whereδ⁰is a normalizing factor defined as δ⁰:= min

w⁰∈W

n

OQ(α, w⁰)+ min

w∈W e∈Evt

TQ(w, e, w⁰)+EQ(e, w)+κ(w) o . Example 4. We continue, using the vocabulary and epistemic state of the previous examples. For illustrative purposes, we keep the observation, transition and event models very simple, with an arbitrary specification: For allw ∈ W, letOQ(β, w) = 0ifw β, else, OQ(β, w) = 1. Let there be two events:Evt ={e1, e2}. LetW = {w1, w2, . . . , wn}.EQ(ek, wi) =i×k, except forEQ(e1, w1) = 0.

TQ(wi, ek, wj) =i×j×k, except forTQ(w1, e1, w1) = 0. Then the ranks of the first two worlds are

• κ_β(qrs) = OQ(β, qrs) + minw∈W e∈Evt

TQ(w, e, qrs) + EQ(e, w)+κ(w) −δ⁰= 0+min{0+0+∞,2+2+3, . . . ,16+

8 + 0} −δ⁰= 7−δ⁰and

• κ_β(qr¯s) = OQ(β, qr¯s) + minw∈W e∈Evt

TQ(w, e, qr¯s) + EQ(e, w) +κ(w) −δ⁰= 1 + min{2 + 1 +∞,4 + 2 + 3,6 + 3 +∞, . . . ,32 + 16 +∞} −δ⁰= 10−δ⁰.

We do not work out the ranks of the other worlds.

Given an epistemic stateκand a new observationα, we propose the following HQBC operation.

κ α:=

(w, n)|w∈W, n= (4)

min{κ^CPIα (w) +es(α, w), κα(w) +os(α, w)} −δ⁰⁰ , whereδ⁰⁰is a normalizing factor defined as

δ⁰⁰:= min

w∈W

min{κ^CPI_α (w) +es(α, w), κ_α(w) +os(α, w)} . κα(w) can be read as ‘The rank ofwafter revision if revision is more plausible given the epistemic strength ofαatw, else, the rank ofwafter update (given the ontic strength).’

4 Analysis of HQBC w.r.t. Rationality Postulates

In this section we shall assess two fundamental postulates generally agreed upon as necessary (but not sufficient) for belief change to be rational [6, e.g.]. Thecategorical matchingpostulate (CM) states that the representation of an agent’s state of knowledge/belief should have the same formal structure before and after the application of the belief change operation under consideration. Thesuccesspostulate (S) states that the observation/evidence with which an agent’s state is to be changed should be believed (with certainty) after the belief change operation.⁸In the rest of this section, we assume thatα∈L is any logically satisfiable piece of information.

8Here it is assumed that the incoming information is certainly correct.

Definition 3. We say

• eventeispossible inκiff there exists a worldw∈W such that κ(w)6=∞andEQ(e, w)6=∞;

• eventeisevent-rationalwhen for allw ∈ W: there exists aw⁰ such thatTQ(w, e, w⁰)6=∞iffEQ(e, w)6=∞;

• evidenceαis ane-signalwhen for allw⁰∈ W: there exists aw such thatTQ(w, e, w⁰)6=∞iffOQ(α, w⁰)6=∞;

• evidenceαistrustworthy iff for allw ∈ W, ifw 6 α, then OQ(α, w) =∞;

• evidenceαiscleariff for allw∈W, ifwα, thenOQ(α, w) = 0;

• evidenceαisweakly observableiff there exists aw ∈ W such thatwαandO(α, w)6=∞;

• evidenceαisstrongly observableiff for allw ∈ W for which wα,O(α, w)6=∞.

Except forpossibility, the definitions in the list above are adapted from Rens [11].

Postulate (CM) Ifκis a ranking function, then so isκ α.

Lemma 1. Ifαis strongly observable, thenκCPIαis a ranking function.

Proof. Omitted to save space; available on request.

Lemma 2. Let the HQBC model be specified such that there exists an event-rational evente∈Evtpossible inκ, andαis ane-signal.

Thenκ αis a ranking function.

Proof. Note that the normalizing factorδwill ensure thatκ αis a ranking function as long as there exists a worldw∈Wfor which κα(w) 6= ∞. It must thus be shown that if there exists an event e^r ∈ Evt which is event-rational andαis ane^r-signal, then there must exist a worldw∈Wfor whichκα(w)6=∞.

κis assumed to be a ranking function. Letw⁻ be a world for whichκ(w⁻) 6= ∞. By definition of the transition function, there must exist a worldw⁺for whichT(w⁻, e, w⁺) 6= ∞, for alle ∈ Evt. Choose thee^rwhich is event-rational. ThenE(e^r, w⁻)6=∞.

Furthermore, becausew⁻exists such thatT(w⁻, e^r, w⁺)6=∞and we know thatαis ane^r-signal,OQ(α, w⁺)6=∞.

By definition of operation (3), κα(w⁺) = OQ(α, w⁺) + minw∈W

e∈Evt

TQ(w, e, w⁺)+EQ(e, w)+κ(w) −δ=OQ(α, w⁺)+

min

. . . , TQ(w⁻, e^r, w⁺) +EQ(e^r, w⁻) +κ(w⁻), . . . −δ 6=

∞.

Proposition 1. If the HQBC model is specified such thatαis strongly observable, there exists an event-rational evente∈Evtpossible in κ, andαis ane-signal, then (CM) holds.

Proof. κ α := {(w, n) | w ∈ W, n = min{κ^CPIα (w) + es(α, w), κα(w) +os(α, w)} −δ⁰⁰}. Recall that neitheres(α, w) noros(α, w) can have a value of∞. And given the antecedents of the proposition, by Lemmata 1 and 2, there must be aw⁰ for whichκ^CPI_α (w⁰) = 0orκ_α(w⁰) = 0. Hence, eitherκ^CPI_α (w⁰) + es(α, w⁰)6=∞orκα(w⁰) +os(α, w⁰)6=∞. Thusκα(w⁰)6=∞ and due to the normalizing factorδ⁰⁰, there exists aws.t.κ_α(w) = 0.

Postulate (S) Ifκis a ranking function, thenκα(α) = 0.

Lemma 3. Ifαis strongly observable, thenκ^CPIα (α) = 0.

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Lemma 4. Let the HQBC model be specified such that there exists an event-rational evente∈Evtpossible inκ, andαis a trustworthy e-signal. Thenκ_α(α) = 0.

Proof. Recall thatκα(α) = minw∈W,wακα(w). By Lemma 2,κα

is a ranking function and thus there exists awfor whichκ_α(w) = 0.

Hence, forκα(α)not to equal 0, there must exist aw⁰ ∈ W s.t.

w⁰ 6αandκ_α(w⁰) 6=∞. But thenOQ(α, w⁰) 6=∞. Therefore, for (S) not to hold, an agent needs to believe thatOQ(α, w⁰)6=∞ for some worldw⁰wherew⁰6α. But thenαcannot be trustworthy.

Arguing by contradiction, (S) must hold.

Note that trustworthiness is required for Lemma 4, in addition to the antecedents required for Lemma 2.

Proposition 2. If the HQBC model is specified such thatαis strongly observable, there exists an event-rational evente∈Evtpossible in κ, andαis a trustworthye-signal, then (S) holds.

Proof. Note that neither es(·) nor os(·) can have a value of ∞.

Moreover, becauseαis trustworthy, by the definitions ofCPIand ,κ^CPIα (w) =κα(w) =∞wheneverw6α. Together with Lem- mata 3 and 4, one can thus infer that

δ⁰⁰= min

w∈W{min{κ^CPIα (w) +es(α, w), κα(w) +os(α, w)}}

= min

w∈W wα

{min{κ^CPI_α (w) +es(α, w), κ_α(w) +os(α, w)}}

Then κ_α(α) = min

w∈W wα

{κ_α(w)}

= min

w∈W wα

{min{κ^CPIα (w) +es(α, w), κα(w) +os(α, w)} −δ⁰⁰}

= min

w∈W wα

{min{κ^CPIα (w) +es(α, w), κ_α(w) +os(α, w)}} −δ⁰⁰

= 0.

5 Comparison of HQBC with Generalized Update

Boutilier [3] adopts an event-based approach where a set of events Eis assumed. These events are allowed to be nondeterministic, and each possible outcome of an event is ranked according to its plausibility via a ranking function. “As in the original event-based seman- tics, we will assume each world has an event ordering associated with it that describes the plausibility of various event occurrences at that world,” [3, p. 14].

Ageneralized update modelis then defined ashW, κ, E, µi, where

• κis a ranking overW (the agent’s epistemic state);

• E is a mapping fromw ∈ W ande ∈ Evt to rankingsκw,e

overW, whereκw,e(w⁰)describes the plausibility that worldw⁰ results when eventeoccurs at worldw;

• µis a mapping fromw ∈ W to rankingsκw overEvt, where κw(e) captures the plausibility of the occurrence of evente at worldw.

In this model, the set of eventsEvt is implicit and the (initial) epistemic stateκexplicit.

Lemma 5. TQ corresponds toE, andEQcorresponds to µ. The correspondence is in the sense that the values of functions are equal for the same arguments, respectively, parameters.

In the rest of the paper, due to Lemma 5, we shall assume that TQ(w, e, w⁰)andκw,e(w⁰)are interchangeable, and thatEQ(e, w) andκw(e)are interchangeable.

Boutilier calls the evolution ofwintow⁰, under evente, atransition, which he writesw→^e w⁰. He defines (rhs in our notation)

κ(w→^e w⁰) :=TQ(w, e, w⁰) +EQ(e, w) +κ(w).

And he defines the set of possibleα-transitions:

Tr(α, κ) :={w→^e w⁰|w, w⁰∈W, e∈Evt, w⁰α, κ(w→^e w⁰)6=∞}.

Tr(α, κ)is the set of transitions from one world to the next via an event, such that the transition (TQ) is possible, the event in the departure world (EQ) is possible, and the departure world (κ) is possible, and such that the arrival world is anα-world.

Then Boutilier defines

result^GU(α, κ) :={w|w⁰→^e w∈minTr(α, κ)}

and definesgeneralized updateas

Bel^GU_α (κ) :={β∈L|result^GU(α, κ)⊆JβK}. (5) Proposition 3. A generalized update model can be realized via an HQBC model.

Proof. LetG=hW, κ, E, µibe a generalized update model, where κ is a (current) epistemic state. Choose an HQBC model H = hW,Evt, TQ, EQ, OQ,osiwith implicit epistemic stateκand such that, for allw, w⁰∈W ande∈Evt,TQ(w, e, w⁰) =κw,e(w⁰)and EQ(e, w) =κw(e). ThenTQcorresponds toEandEQcorresponds toµ.Gis thus realized viaH.

Lemma 6. LetHbe the class of HQBC models specified such that there exists an event-rational event e ∈ Evt possible in κ and such that evidenceαis ane-signal, trustworthy and clear. For every generalized update model realizable via an HQBC model inH, Bel^GUα (κ) =Bel(κα).

Proof. LetH ∈ Hs.t.H = hW,Evt, TQ, EQ, OQ,osiwith implicit epistemic state κ. Let G = hW, κ, E, µi be a generalized update model realized via H. Note thatκ(w →^e w⁰) 6= ∞ iff TQ(w, e, w⁰)6=∞, EQ(e, w)6=∞andκ(w)6=∞.

Bel^GU_α (κ) =Bel(κ_α)iff{β ∈ L |result^GU(α, κ) ⊆JβK} = {β ∈L | (κα)⁻¹(0)⊆JβK}(by their definitions: (5) and (2)) iff result^GU(α, κ) = (κ_α)⁻¹(0)iffresult^GU(α, κ) =JBel(κ_α)K.

Andresult^GU(α, κ) =

{w|w⁰→^e w∈minTr(α, κ)}(by definition ofresult^GU)

={w|w⁰→^e w∈min{w⁰→^e w|w⁰, w∈W,

e∈Evt, wα, κ(w⁰→^e w)6=∞}}(by definition ofTr)

= arg min

w:

w,w⁰∈W, e∈Evt, wα T_Q(w⁰,e,w)+E_Q(e,w⁰)

+κ(w⁰)6=∞

{TQ(w⁰, e, w) +EQ(e, w⁰) +κ(w⁰)}

(by definition ofκ(w⁰→^e w))

= arg min

w∈W

OQ(α, w)

+ min

w⁰∈W,e∈Evt{TQ(w⁰, e, w) +EQ(e, w⁰) +κ(w⁰)}

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(by definition of classH,wαandTQ(w⁰, e, w) +EQ(e, w⁰) + κ(w⁰)6=∞)

= n

arg min

w∈W

{κα(w)}o

= (κα(w))⁻¹(0) =JBel(κα)K.

Proposition 4. LetHbe the class of HQBC models specified such that there exists an event-rational evente ∈ Evt possible inκand such that evidenceαis ane-signal, trustworthy and clear. For every generalized update model realizable via an HQBC model inH, Bel^GUα (κ) =Bel(κα).

Proof. Letos(α, w) = 0for all possibleαand for allw∈W. Let τ > κα(w)−κ^OGIα (w)for allw ∈ W for whichκα(w) 6= ∞ andκ^OGI_α (w)6=∞. Recall thatτmay not equal∞and for allw∈ W,os(α, w)6=∞.

Then, for allw∈W,

κ_α(w) = min{κ^OGIα (w) +es(α, w), κ_α(w) +os(α, w)} −δ⁰⁰

= min{κ^OGIα (w) +τ−os(α, w), κα(w) +os(α, w)} −δ⁰⁰ (by Def. 2)

= min{κ^OGI_α (w) +τ, κ_α(w) + 2os(α, w)} −δ⁰⁰

= min{κ^OGIα (w) +τ, κα(w)} −δ⁰⁰ (by the above definition ofos(α, w))

=κ_α(w)−δ⁰⁰(by the above definition ofτ)

=κ_α(w)(by definition, min

w⁰∈Wκ_α(w⁰) = 0).

Then Bel(κα) = Bel(κα) and by Lemma 6, Bel(κα) = Bel^GU_α (κ).

6 Examples

We use Boutilier’s two examples [3,§3.3]. One can then compare hisgeneralized update(GU) with our HQBC.

The first example involves a book (B) which might be inside the house or on the patio. There are three events: it rains, in which case the grass (G) and the patio get wet, the sprinkler comes on, in which case only the grass gets wet, or nothing happens. In this example, events are deterministic. If the book is on the patio, it will get wet when it rains, else not. If the book is inside and the book is dry, it will never get wet. Figure 1 illustrates the prior epistemic state of an agent who believes its book is on the patio and that both the grass and the book are dry (κ(Patio(B)∧Dry(B)∧Dry(G)) = 0), but if the book is not on the patio, the agent believes it has left it inside (κ(Inside(B)∧Dry(B)∧Dry(G)) = 1). The other less plausible worlds are omitted. Event plausibility is ranked asEQ(null, w) = 0, EQ(rain, w) = 1,EQ(sprinkler, w) = 2, for allw(a ‘global’ ordering suitable for all worlds is assumed). This is the only information required for GU.

For HQBC, the observation function (OQ), ontic strength (os) and its top value (τ), and distance measure (d) are required, in addition. We let all observations be trustworthy and clear. For now, let the agent have no opinion as to whether observations are ontic or epistemic, that is, for all possible α and for all w ∈ W, os(α, w) = es(α, w) = 1.⁹ dwill be defined in accordance with Hamming distance, as before.

9This implies thatτ= 2.

Figure 1: Scenario with multiple events (with deterministic out- comes), including event plausibility information.

Suppose the agent observes that the grass is wet (¬Dry(G)). This contradicts what the agent presently believes, so, with respect to revision,κCPI¬Dry(G)is interpreted asκOGI¬Dry(G). We determine that

• κ^OGI_¬Dry(G)(Patio(B)∧Dry(B)∧ ¬Dry(G)) = 0

• κ^OGI_¬Dry(G)(¬Patio(B)∧Dry(B)∧ ¬Dry(G)) = 1

• κ^OGI_¬Dry(G)(w) =∞for allw∈Ws.t.w6Patio(B)∧Dry(B)∧

¬Dry(G)andw6¬Patio(B)∧Dry(B)∧ ¬Dry(G) With respect to update, we determine that

• κ¬Dry(G)(Patio(B)∧Dry(B)∧ ¬Dry(G)) = 1

• κ_¬Dry(G)(Patio(B)∧ ¬Dry(B)∧ ¬Dry(G)) = 0

• κ_¬Dry(G)(w) =∞for allw∈Ws.t.w6Patio(B)∧Dry(B)∧

¬Dry(G)andw6Patio(B)∧ ¬Dry(B)∧ ¬Dry(G) Then combining these results gives

• κ_¬Dry(G)(Patio(B)∧Dry(B)∧ ¬Dry(G)) = 0

• κ_¬Dry(G)(Patio(B)∧ ¬Dry(B)∧ ¬Dry(G)) = 0

• κ_¬Dry(G)(¬Patio(B)∧Dry(B)∧ ¬Dry(G)) = 1

and the other worlds are deemed impossible. If the agent were to reflect on its new beliefs, it might reason as follows.

I believePatio(B)∧Dry(B)∧ ¬Dry(G)because it is the

¬Dry(G)-world closest to my prior beliefs (and at least plausible, because it is plausibly explained by the sprinkler coming on). I believePatio(B)∧ ¬Dry(B)∧ ¬Dry(G)because it is a¬Dry(G)-world best explained by rain (in my prior beliefs).

I don’t fully believe¬Patio(B)∧Dry(B)∧ ¬Dry(G), but it is plausible because it is the¬Dry(G)-world second closest to my prior beliefs (although¬Patio(B)was previously not fully believed, it was deemed plausible.).

Now suppose the ontic strength of ¬Dry(G) is defined as os(¬Dry(G), w) = 0, for allw ∈ W, andτ = 2. That is, perceiving wet grass is always deemed slightly more ontic than epistemic. Then the resulting epistemic state is determined as in Ta- ble 1. In the table, worlds are identified by three letters, such that, for instance,pdd |=Patio(B)∧Dry(B)∧Dry(G)andiww |=

¬Patio(B)∧ ¬Dry(B)∧ ¬Dry(G);OGI abbreviates κ^OGI_α (w), abbreviatesκ_α(w),es abbreviateses(α, w) andos abbreviated

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os(α, w), whereαis the incumbent observation andwis the world of the row. The “min” column indicates the minimum value between the two columns to its left, and is actually the rank assigned to the worldwof the incumbent row (κα(w)).

Table 1: Agent prefers an ontic interpretation.

World OGI+es +os min pdd ∞+ 2 ∞+ 0 ∞

pdw 0 + 2 1 + 0 1

pwd ∞+ 2 ∞+ 0 ∞

pww ∞+ 2 0 + 0 0

idd ∞+ 2 ∞+ 0 ∞

idw 1 + 2 ∞+ 0 3

iwd ∞+ 2 ∞+ 0 ∞ iww ∞+ 2 ∞+ 0 ∞

Finally, suppose ontic strength of ¬Dry(G) is defined as os(¬Dry(G), w) = 2for allw ∈W, withτ = 2(which implies thates(¬Dry(G), w) = 0, for allw∈W). That is, perceiving wet grass is always deemed more epistemic than ontic. Then the resulting epistemic state is determined as in Table 2.

Table 2: Agent prefers an epistemic interpretation.

World OGI+es +os min pdd ∞+ 0 ∞+ 2 ∞

pdw 0 + 0 1 + 2 0

pwd ∞+ 0 ∞+ 2 ∞

pww ∞+ 0 0 + 2 2

idd ∞+ 0 ∞+ 2 ∞

idw 1 + 0 ∞+ 2 1

iwd ∞+ 0 ∞+ 2 ∞ iww ∞+ 0 ∞+ 2 ∞

We now analyze the results of the two tables/cases a little.

We see that when the agent prefers to interpret or explain

¬Dry(G)as an ontic observation, the agent considers worldpww as most plausible, that is, it fully believes that the book is on the patio, the book is wet and the grass is wet. A reason could be that, given the agent’s most plausible prior belief that the book is on the patio and dry and the grass is dry, it rained. This is the same result pro- duced by generalized update [3, p. 17]; this correspondence makes sense, given that our update () is ‘aligned’ with GU (Bel^GUα (κ)

=Bel(κα) under reasonable conditions; Lem. 6). Notice that the plausibility of pww due to revision is not in contention, because κ^OGI_¬Dry(G)(pww) =∞.

We see that when the agent prefers to interpret or explain

¬Dry(G) as an epistemic observation, the agent considers world pdwas most plausible, that is, it fully believes that the book is on the patio, the book is dry and the grass is wet. It can be seen from Table 2 that it is revision which causes the agent to believepdw. No- tice thatpdwis the Hamming-closest¬Dry(G)-world to the most plausible prior belief (pdd).

The second example is shown in Figure 2. Here only one possible event is assumed, the action of dipping litmus paper in a beaker.

The beaker is believed to contain either an acid or a base (κ = 0); little plausibility (κ = r) is accorded the possibil- ity that it contains some other substance (say, kryptonite). The expected outcome of the test is a color change of the litmus paper: it changes from yellow to red if the substance is an acid, to blue if it is a base, and to green if it is kryptonite. However, the litmus test can fail some small percentage of the time, in which case the paper also turns green. This outcome is also accorded little plausibility (κ = g). If the paper is dipped, and redis observed, the agent will adopt the new beliefacid. Un- like KM update [of Katsuno and Mendelzon [8]], generalized

Figure 2: Scenario with single event (with non-deterministic out- comes), including event plausibility information.

update permits observations to rule out possible transitions, or previously epistemically possible worlds. As such, it is an appropriate model for revision and expansion of beliefs due to information-gathering actions. An observed outcome ofgreen presents two competing explanations: either the test failed (the substance is an acid or a base, and we still dont know which) or the beaker contains kryptonite. The most plausible explanation and the updated epistemic state depend on the relative magni- tudes ofgandr. The figure suggests thatg < r, so the a test failure is most plausible and the beliefacid∨baseis retained.

If test failures are more rare (r < g), then this outcome would cause the agent to believe the beaker held kryptonite. [3, p. 18]

Now we investigate how HQBC deals with this scenario for two observations. We let all observations be trustworthy and clear, and Hamming distance is used to defined. The three possible observations arered,blueandgreen.

Tables 3 and 4 show the agent’s new epistemic state after perceiving the litmus paper turning red, respectively, green. In both tables, the three right-most columns report the new state (κα) when the agent (from left to right) (i) is indifferent about whether the observation is ontic or epistemic, (ii) prefers an ontic interpretation, (iii) prefers an epistemic interpretation. “os = x” (“es =x”) in a column heading means thatos(α, w) =x(resp.,es(α, w) =x) for all w∈W. In the tables, worlds are identified by two letters, such that, for instance,ar|=acid∧red,ab|=acid∧blue,bg|=base∧green, ky |=krypt∧yellow. To save space, rows containing∞in every row are omitted. Of course, perceiving red, blue or green is inconsistent with the current belief that the litmus paper is yellow; revision operatorCPIis thus interpreted asOGI.

Table 3: Agent perceives the litmus paper turning red.

World OGI es = 1

os = 1

es = 2

os = 0

es = 0

os = 2

ar 0 0 0 0 0

br 0 ∞ 0 2 0

kr r ∞ r r+ 2 r

We see that when the agent prefers to revise its beliefs (es = 0, os= 2), and when it is indifferent about whether to revise or update (es = 1,os = 1), then its resulting beliefs seem unintuitive to us humans—the agent believes as equally plausible that the substance is acid and that it is base. However, when the agent prefers to update its beliefs (es = 2,os = 0), then it reasonably believes (only) that the substance is acid. A reasonable agentshouldprefers to update its

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beliefs because it should consider all its observations in this scenario to be ontic, due to the ontic nature of dipping litmus paper.

Table 4: Agent perceives the litmus paper turning green.

World OGI es = 1

os = 1

es = 2

os = 0

es = 0

os = 2

ag 0 g 0 0 0

bg 0 g 0 0 0

kg r r+r r r r

Note that for the case whenes= 2andos= 0, the values had to be normalized (δ⁰⁰= 2). It is interesting to see that no matter what stance the agent takes on ontic/epistemic strength, when it perceives green, it believes with equal plausibility that the substance is acid and base. Assumingr >0, the substance is less plausibly kryptonite, but not impossible.

In the cases when the agent has an event-based attitude (i.e., it prefers to interpret observations ontically), all results when HQBC is applied align with the results when GU is applied (w.r.t. the examples in this paper).

7 Concluding Remarks, Related and Future Work

A hybrid qualitative belief change (HQBC) construction was presented—based on ranking theory and which trades revision off with update, according to the agent’s confidence for whether the received observation/evidence is ontic (due to a physical event) or epistemic (due to an announcement). We proved that HQBC is, in a particular sense, an extension of Boutilier’s generalized update (GU). In other words, the HQBC model in a class of ‘reasonable’ models can be specified to perform exactly the same belief change as GU would.

Moreover, HQBC allows for more sophisticated belief change than GU, in particular, with respect to rankic belief revision (based on conditional plausibility and generalized imaging, for instance) and with respect to employing a notion of ontic strength. The examples in this paper support our propositions concerning the relationship between HQBC and GU.

Determiningos(α, w)for every foreseenαin every possible worldwwill be challenging for a designer. Some deep questions are: Should the designer/agent provide the strengths (via stored values or programmed reasoning), or do these strengths come to the agent attached to the new information? What is the reasoning process we go through to determine whether information is epistemic or ontic, if at all? In general, how does an agent know when information is epistemic (requiring revision) or ontic (requiring update)? [11]

One direction to investigate as a possible answer to the questions above is to condition ontic/epistemic strength on particular propositions. For instance, the more plausible the proposition, the more likely that the received information is ontic. For such an approach to work, the framework would presumably have to accommodate the specification ofcondition propositionsfor every observation of in- terest. Revision and update would then be traded off depending on the plausibility/probability of the condition of the observation under consideration.

Friedman and Halpern [5] investigate belief revision and update employing a framework based on time-stamps and runs of possible evolutions of a system. They provide some interesting insights re- garding the relationship between revision and update, which may

also benefit our future work. In their concluding section, they hint that their framework could ‘mix’ revision and update: “In this framework, belief change operations can be determined by choosing a plausibility measure that captures the agent’s preferences among se- quences of worlds.... [T]here are prior plausibilities that, when con- ditioned on a surprising observation, allow the agent to revise some earlier beliefs and to assume that some change has occurred”, [5]. To our knowledge, they never did investigate the ‘mixture’ approach.

Relationships to the change operations defined by Beierle and Kern-Isberner [1, 2], which make use of knowledge bases, also need to be investigated.

Although Lang’s work [?] is not directly applicable to ours in terms of ‘mixing’ revision and update, it does unpack and high- light several important characteristics of update. Lang also discusses the relationship between update to revision. His insights might well guide our future efforts in this area. He writes

In complex environments, especially planning under in- complete knowledge, actions are complex and have both ontic and epistemic effects; the belief change process then is very much like the feedback loop in partially observable planning and control theory: perform an action, project its effects on the current epistemic state, then get the feedback, and revise the projected epistemic state by the feedback. Clearly, update allows for projection only. Or, equivalently, if one chooses to separate the ontic and the epistemic effects of actions, by hav- ing two disjoint sets of actions (ontic and epistemic), then ontic actions lead to projection only, while epistemic actions lead to revision only. Therefore, if one wants to extend belief update so as to handle feedback, there is no choice but integrating some kind of revision process, as in several recent works [. . . ] [?]

This act-update-perceive-revise “feedback loop” is the default approach when complex actions/events are considered; it is fundamen- tally different to the simultaneous, hybrid belief change approach.

Yet, we have not come across a convincing argument against the hybrid approach. It seems that the traditional “feedback loop” approach assumes that there is always certainty about the ontic/epistemic status of every piece of information received. A major question for future research is, Is there a theory or framework to synthesize the two approaches?

Nayak [10] proves that, given an appropriate function for measur- ing distance between worlds, classical revision (∗) can be reduced to classical update (). Formally, he proves that(x∗k)x=k∗x, wherek, x∈L,kis an agent’s knowledge andxis the (new) evidence. Nayak points out that the “nice storyline that cleanly demar- cates revision from update appears not to be such a good story after all,” [10]. Nayak’s surprising result is just one more reason to investigate the hybrid belief change approaches.

“We can regard imaging as a probabilistic version of update, and conditionalization as a probabilistic version of revision,” [8]. And according to Nayak, KM-update “is known to be the non-probabilistic counterpart of the account of [probabilistic] imaging propounded by David Lewis in order to develop a theory of conditionals [. . . ]” [10].

Dubois and Prade [4] give a version of imaging for belief update in the possibilistic framework. Rens [11] uses imaging (GI) on the revision side; his justification is because imaging can deal with contra- dictory evidence, whereas conditioning cannot. We are not convinced that imaging is strictly an update process. Where exactly imaging lies on the revision-update spectrum is, to our minds, another deep question still to be answered.

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